Solve the inequalities. Where appropriate, give an exact answer as well as a decimal approximati

step 1 So the inequality we have to solve is 10 to the power minus x square is less than equal to 10 to

Solve the inequalities. Where appropriate, give an exact...

Key Concepts

Quadratic Inequalities

Once an inequality in exponential form is reduced to an inequality involving the exponent, it may lead to a quadratic inequality such as x^2 ? constant. Solving quadratic inequalities involves finding the values of x for which the squared term exceeds the constant. This typically results in a solution that is expressed in terms of square roots, requiring both an exact answer (in radical form) and possibly a decimal approximation for clarity. Understanding how to handle quadratic inequalities is fundamental in many areas of algebra.

Exponential Functions and Monotonicity

Exponential functions, such as the function with base 10, are critical in understanding inequalities of the form 10^a ? 10^b. Since the exponential function f(x) = 10^x is strictly increasing, comparing the values of the function can be reduced to comparing the exponents. This means that if 10^a ? 10^b holds, then it must be that a ? b. This property is instrumental in solving inequalities where the exponential form appears on both sides with the same base.

Inequality Transformation and Sign Reversal

When manipulating inequalities, especially those involving exponents, it is often necessary to perform algebraic operations that may involve multiplying or dividing by negative numbers. This operation requires careful attention because multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality. In this context, transforming the exponent inequality often involves such a sign change, which is crucial for arriving at the correct solution.

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